MIDI notes cutoff

Hi 
I still use pro audio 9 and lose notes when I paste from the exact start of a bar.  Is there an option in Sonar (or pro audio) that will tack on the previous quarter (or eighth) of a beat of MIDI information w/o having to actually paste from the previous quarter?  Obviously some notes are going to be slightly ahead of the beat occasionally.  It's difficult to find search terms that narrow this down.  
 
Thanks

Snap Offset won't be helpful for copying/moving partial clip selections as I gather is the case here.
 
Unfortunately, there is no longer a 'fuzzy' selection option as there was way back in the early days of Cakewalk for DOS when there was no quantizing, and you could only copy entire measures.
 
I suggest you set snap at a value that's big enough to encompass early notes when drag-selecting, but small enough not to accidentally include notes that belong in the previous measure (maybe 1/16), and then use drag-copy/move with Snap 'By' enabled instead of cut/copy-paste to preserve the tick offset of the selection. I think this would be applicable in Pro Audio.
 
In SONAR you can also select in the PRV by lassoing (right-click and drag across a time and pitch range) which will select entire notes by just touching them, but I don't think Pro Audio had a Lasso function.

Thanks for the previous replies.
Today I have been punching in by using the staff view.  I can select the resolution down to 16th notes and then Shift/left arrow to scroll through backwards by that small increment.  So I autopunch in and out 1/16th early.  
Obviously, this doesn't work for cut/paste drag/drop.  That's probably best as I get a little lazy just re-using exact sections!  
Thanks again